NATO was established as a western European counterpoint to the Warsaw Pact, dominated by the USSR. In Fallout's case, China is 'the other', not the USSR, so NATO would have waning relevance.
If NATO did exist (as it probably did at some point in the fallout world, as it was created in 1949), my understandng is the basic idea with NATO is an attack on one would be considered an attack on all.
Edit: Quoting Wikipedia, which quotes the treaty:
The Parties of NATO agreed that an armed attack against one or more of them in Europe or North America shall be considered an attack against them all. Consequently they agree that, if such an armed attack occurs, each of them, in exercise of the right of individual or collective self-defence will assist the Party or Parties being attacked, individually and in concert with the other Parties, such action as it deems necessary, including the use of armed force, to restore and maintain the security of the North Atlantic area
Assuming the treaty was still in force, China invading the US would therefore be considered an attack on all NATO countries - Whats left the European allies, and also Canada. Since Canada showed some reluctance to allow troops to pass through its territory before 2076, I think its unlikely that NATO existed at this point.
If it did, in 2076 it would have had to have broken down - The US Annexed Canada, itself an act of war, in theory forcing the rest of NATO to declare war on the US.